Generalized urn models of evolutionary processes. (Q1879916)

The authors study a class of urn models which generalizes the one introduced in the article by \textit{S. J. Schreiber} [SIAM J. Appl. Math. 61, 2148--2167 (2001; Zbl 0993.92026)]. Models of this type can be used to describe finite populations; three examples of such applications are considered in the paper -- the replicator process and fertility-selection process with and without mutation. (These processes are stochastic versions of the replicator equations and fertility-selection equations of population genetics.) The authors investigate the limiting behavior of the generalized urn process (GUP). To do that, they introduce the mean limit ODE and use a theorem relating the asymptotic properties of its solution to those of the GUP proved by Schreiber [loc. cit.]. In particular, this approach allows them to carry over the exclusion and time averaging principles, which hold for the replicator equation, to the replicator process. It is shown that when the limit ODE admits an attractor at which growth is expected, the population grows with positive probability and its genotypic composition converges to the attractor with positive probability. Also, the authors describe two types of invariant sets of the mean limit ODE to which the GUP does not converge. Finally, they establish conditions for growth with positive probability for processes with gradient-like mean limit ODEs.